SEMINAR ON EIGENVALUES SEMINAR ON EIGENVALUES AND EIGENVECTORSAND EIGENVECTORS

By Vinod Srivastava M.E. Modular I & CRoll No.151522

CONTENTSCONTENTS Introduction to Eigenvalues and Eigenvectors Examples Two-dimensional matrix Three-dimensional matrix• Example using MATLAB• References

INTRODUCTIONINTRODUCTION

Eigen Vector-

In linear algebra , an eigenvector or characteristic vector of a square matrix is a vector that does not changes its direction under the associated linear transformation.

In other words – If V is a vector that is not zero, than it is an eigenvector of a square matrix A if Av is a scalar multiple of v. This condition should be written as the equation:

AV= λv

Contd….Contd….

Eigen Value-• In above equation λ is a scalar known as the eigenvalue or

characteristic value associated with eigenvector v.

• We can find the eigenvalues by determining the roots of the characteristic equation-

0 IA

ExamplesExamples

Two-dimensional matrix example-Ex.1 Find the eigenvalues and eigenvectors of matrix A.

Taking the determinant to find characteristic polynomial A-

 It has roots at λ = 1 and λ = 3, which are the two eigenvalues of A.

2112

A

0IA 021

12

043 2

Eigenvectors v of this transformation satisfy the equation, Av= λvRearrange this equation to obtain-

For λ = 1, Equation becomes,

which has the solution,

0 vIA

0 vIA

00

1111

2

1

vv

1

1v

For λ = 3, Equation becomes,

which has the solution-

Thus, the vectors vλ=1 and vλ=3 are eigenvectors of A associated with the eigenvalues  λ = 1 and λ = 3, respectively.

03 uIA

00

1111

2

1

uu

11

u

Three-dimensional matrix example-Ex.2 Find the eigenvalue and eigenvector of matrix A.

the matrix has the characteristics equation-

200130014

A

0234

200130

014

AI

therefore the eigen values of A are-

For λ = -2, Equation becomes,

which has the solution-

4,3,2 321

000

000110

012

0

3

2

1

11

vvv

vAI

221

v

Similarly for λ = -3 and λ = -4 the corresponding eigenvectors u and x are-

002

,011

xu

Example using MATLABExample using MATLAB

REFERENCESREFERENCES http://www.slideshare.net/shimireji Digital Control and State Variable methods by M.Gopal https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

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